System and method for improving the minimization of the interest rate risk

ABSTRACT

A system for interest rate risk management, comprising: an input device, configured to receive as input a first group of data indicative of a first group of financial instruments to be protected; a second group of data indicative of a second group of financial instruments aimed at protecting said first group of financial instruments; and an interest rate risk minimization device, connected to said input device, configured to receive as input said first and second group of data, a data feed of current market prices of said first and second group of financial instruments, a set of parameters of a term structure model, and historical zero-coupon term structures of interest rates, and to generate historical model errors and, considering these errors, the optimal amount to be invested in each financial instrument which shall be used to protect the balance sheet or portfolio. The interest rate risk minimization device is further configured to generate a residual risk estimation.

This is a continuation-in-part of U.S. Ser. No. 12/461,360 filed on Aug.10, 2009

The present invention relates generally to risk management. Morespecifically, the present invention relates to systems and methods forimproving interest rate risk management for institutional investors,like banks, insurance companies and pension funds, and portfoliomanagers.

BACKGROUND OF THE INVENTION

The level of interest in Liability Driven Investments (LDI) and, moregenerally, in accurate techniques of asset and liability management hasgrown up significantly over the last decade.

This follows a process of de-risking which has been implementedworldwide by many institutional investors.

Accordingly, the approaches to effectively hedge against interest raterisk have become significantly more sophisticated than the initialmodels based on duration and convexity.

Hedging interest rate risk relies on approximating the dynamics of theterm structure of interest rates, that is the relationship between thelevel of a certain interest rate and its maturity, through a modelconsidering a limited number of factors. This leads to a differencebetween the modeled and the actual dynamics of interest rates, which wewill define as the model error.

The theories underpinning these approaches mostly rely on the conceptsof key rate duration (KRD), of duration vectors (DV), generalizedduration vectors (GDV) or on Principal Component Analysis (PCA).

Hedging based on PCA is one of the most common techniques used byinstitutional investors to minimize the basis risk from shifts in theyield curve. For this reason, this description of the invention is basedon a PCA model. However, the key ideas of the invention can be appliedalso to key rate duration, duration vectors or M-vector models.

In the classical Asset Liability Management interest rate riskmanagement problem, a portfolio of liabilities V is given, which at timet has a value of V_(t) and with cash flows that are grouped in m timebuckets. The present value of the liabilities included in the i-th timebucket amounts is A_(i). In a context of portfolio management, portfolioV would simply represent the portfolio of investments which must behedged.

For each of these time buckets, basis risk comes from unexpected shiftsin the corresponding zero-coupon risk free rate R(t,D_(k)), where D_(k)indicates the duration and maturity of the time bucket.

In order to immunize the portfolio of liabilities within the followingexample, a hedging portfolio H is used and is composed of several couponbonds y with cash flows that are grouped in n time buckets. Thepercentage of the present value of bond y represented by the cash flowwith maturity D_(k) is indicated by ω_(y),k. Alternatively, theportfolio H could include other financial instruments, like interestrate swaps, bond futures, or interest rate futures.

The optimal amount to be invested in a specific coupon bond y isindicated by Φ_(y), which is the final value to obtain by the resolutionof the problem.

Usually hedging strategies assume the so-called self-financingconstraint:

$\begin{matrix}{{\sum\limits_{y = 1}^{M + 1}\varphi_{y,t}} = {H_{t} = V_{t}}} & (2.)\end{matrix}$

In PCA-hedging models as used in the prior art, the set of equations tosolve in order to obtain the optimal weights Φy is defined as follows:

$\begin{matrix}{{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}{\sum\limits_{k = 1}^{n}{c_{lk}D_{k}\omega_{y,k,t}}}}} = {\sum\limits_{k = 1}^{m}{c_{lk}D_{k}A_{k,t}}}} & (5.)\end{matrix}$

where c_(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to the change in the l-th principal component and Mrepresents the number of considered principal components and normally istwo or three. Obviously, equation (5) would need to be adjusted if otherfinancial instruments, like interest rate swaps, bond futures, orinterest rate futures are included within the hedging portfolio H. Theseadjustments rely on well-known techniques.

This equation must be true for each principal component l. Equation (5.)ensures that the sensitivity of the two portfolios V and H to thedynamics of each principal component is equal.

In theory, PCA with three principal components (traditionally identifiedas the level, the steepness, and the curvature of the term structure)should improve the quality of hedging, since it allows to hedge alsoagainst changes in the curvature of the yield curve. However, empiricalevidence exists suggesting that hedging based on PCA should rely ratheron two principal components than on three.

Additional empirical evidence suggests that also other models whichshould in theory allow to better capture the dynamics of the yield curvedo not necessarily lead to better hedging.

In KRD-hedging models as used in the prior art, the set of equations tosolve in order to obtain the optimal weights Φy is defined as follows:

$\begin{matrix}{{\sum\limits_{k = 1}^{n}{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}c_{l,k}^{KRD}D_{k}}}} = {\sum\limits_{k = 1}^{m}{A_{k,t}c_{l,k}^{KRD}D_{k}}}} & \left( {5.a} \right)\end{matrix}$

where c^(KRD) _(l,k) represents the sensitivity of the zero-coupon rateof maturity D_(k) to the change in the l-th key rate and M representsthe number of considered key rates and normally is two or three. Thisequation must be true for each key rate l. Equation (5.a) ensures thatthe sensitivity of the two portfolios V and H to the dynamics of eachkey rate is equal.

A third approach relies on GDV-hedging models as used in the prior art,where the set of equations to solve in order to obtain the optimalweights Φy is defined as follows:

$\begin{matrix}{{\sum\limits_{k = 1}^{n}{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}D_{k}^{\chi {(l)}}}}} = {\sum\limits_{k = 1}^{m}{A_{k,t}D_{k}^{\chi {(l)}}}}} & \left( {5.b} \right)\end{matrix}$

where D_(k) ^(X(l)-l) represents the sensitivity of the zero-coupon rateof maturity D_(k) to the change in the l-th risk factor and M representsthe number of considered risk factors and normally is two or three. Thisequation must be true for each risk factor l. Equation (5.b) ensuresthat the sensitivity of the two portfolios V and H to the dynamics ofeach risk factor is equal.

The DV approach is a special case of the GDV approach where X(l) hasbeen set equal to l.

The size of the hedging error is influenced by the interaction of twomain factors: the model error, that is the difference between themodeled and the actual dynamics of the yield curve, and the level ofexposure of the overall portfolio, represented by the sum of the assetsand the liabilities, to the model errors.

A higher exposure to model errors could outbalance the positive effectof a more sophisticated yield curve model capable of reducing the sizeof these errors.

In the state of art, traditional hedging based on PCA or on other modelsdoes not control the level of exposure to the model errors.

SUMMARY OF THE INVENTION

The aim of the present invention is to provide a new system and methodfor interest rate risk management overcoming the above mentioneddrawbacks.

Within this aim, an object of the present invention is to introduce ageneralized hedging model able to control the overall exposure to themodel errors and reduce residual interest rate risk.

Another object of the invention is to provide a rigorous estimate ofresidual risk.

This aim and other objects which will become better apparent hereinafterare achieved by a system for interest rate risk management (100),comprising:

-   -   an input device, configured to receive as input a first group of        data indicative of a first group of financial instruments to be        protected; a second group of data indicative of a second group        of financial instruments for protecting said first group of        financial instruments; and    -   an interest rate risk minimization device, connected to said        input device, configured to receive as input said first and        second group of data, a data feed of current market prices of        said first and second group of financial instruments, a set of        parameters of a term structure model, and historical zero-coupon        termstructures of interest rates;        wherein said interest rate risk minimization device is further        configured to generate: historical model errors;    -   an optimal amount (Φ_(y)), obtained as a function of said        historical model errors, for investment in each financial        instrument of said second group of financial instruments for        protecting said first group of financial instruments; and a        residual risk estimation (E[Ψ_(t) ²]);        wherein said interest rate risk minimization device is further        configured to solve equation:

${\frac{\partial{L\left( {\phi_{t},\mu_{t}} \right)}}{\partial\varphi_{y,t}} \approx \approx {{2{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{{\left\lbrack {\sum\limits_{k = 1}^{n}{c_{lk}D_{k}w_{y,k,t}}} \right\rbrack\left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left( {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right)}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{\theta_{k}^{2}c_{lk}^{2}D_{k}^{2}{w_{y,k,t}\left\lbrack {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right\rbrack}}}} \right\}}}} - \mu_{t}}} = 0$

to obtain μ_(t) and said optimal amount (Φ_(y)),wherein D_(k) is a zero-coupon rate of maturity, C^(l) _(t) representsthe change in the l-th principal component between time t and t+1,c_(lk) represents the sensitivity of the zero-coupon rate of maturityD_(k) to this change, said C^(l) _(t) and c_(lk) being the set ofparameters of a term structure model, w_(y,k,t) and w_(j,k,t), indicatethe percentage of the present value of bond y and j, respectively,represented by the cash flow with maturity D_(k,) _(A) _(,k,t) is thevalue of the liabilities included in the k-th time bucket amounts, μ_(t)is the Lagrange multiplier andwherein the ratio θ_(k), the volatility σ_(C) of the modeled rate shiftsand the model errors σ_(ε) are defined as

${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$

Another aim and other objects which will become better apparenthereinafter are achieved by a system for interest rate risk management(100), comprising:

-   -   an input device, configured to receive as input a first group of        data indicative of a first group of financial instruments to be        protected; a second group of data indicative of a second group        of financial instruments for protecting said first group of        financial instruments; and    -   an interest rate risk minimization device, connected to said        input device, configured to receive as input said first and        second group of data, a data feed of current market prices of        said first and second group of financial instruments, a set of        parameters of a term structure model, and historical zero-coupon        termstructures of interest rates;        wherein said interest rate risk minimization device is further        configured to generate: historical model errors;    -   an optimal amount (Φ_(y)), obtained as a function of said        historical model errors, for investment in each financial        instrument of said second group of financial instruments for        protecting said first group of financial instruments; and a        residual risk estimation (E[Ψ_(t) ²]);        wherein said interest rate risk minimization device is further        configured to solve the following equation for each bond j        included in the hedging portfolio to obtain μ_(t) and said        optimal amount (Φ_(y)):

$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$

wherein the factor F^(l) represents the l-th key zero rate change,c^(KRD) _(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to this change, ω_(y,k,t) and ω_(j,k,t), indicate thepercentage of the present value of bond y and j, respectively,represented by the cash flow with maturity D_(k,) _(A) _(,k,t) is thevalue of the liabilities included in the k-th time bucket amounts, μ_(t)is the Lagrange multiplier andthe volatility of the model errors σ_(ε) can be defined as in equation

${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$

Another aim and other objects which will become better apparenthereinafter are achieved by a system for interest rate risk management(100), comprising:

-   -   an input device, configured to receive as input a first group of        data indicative of a first group of financial instruments to be        protected; a second group of data indicative of a second group        of financial instruments for protecting said first group of        financial instruments; and    -   an interest rate risk minimization device, connected to said        input device, configured to receive as input said first and        second group of data, a data feed of current market prices of        said first and second group of financial instruments, a set of        parameters of a term structure model, and historical zero-coupon        term structures of interest rates;        wherein said interest rate risk minimization device is further        configured to generate: historical model errors;    -   an optimal amount (Φ_(y)), obtained as a function of said        historical model errors, for investment in each financial        instrument of said second group of financial instruments for        protecting said first group of financial instruments; and a        residual risk estimation (E[Ψ_(t) ²]);        wherein said interest rate risk minimization device is further        configured to solve the following equation for each bond j        included in the hedging portfolio to obtain μ_(t) and said        optimal amount (Φ_(y)):

$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - t}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$

wherein F^(l) represents the l-th risk factor change, D_(k) ^(X(l)-l)represents the sensitivity of the zero-coupon rate of maturity D_(k) tothis change, ω_(y,k,t) and ω_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and the volatility of the model errors σ_(ε) can be definedas in equation

${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$

Another aim and other objects which will become better apparenthereinafter are achieved by a method for interest rate risk management,comprising:

-   -   receiving as input into an input device, a first group of data        about a first group of financial instruments to be protected and        a second group of data about a second group of financial        instruments protecting said first group of financial        instruments;    -   receiving as input into an interest rate risk minimization        device connected to said input device: said first and second        group of data; a data feed of current market prices of said        first and second group of financial instruments; a set of        parameters of a term structure model; and historical zero-coupon        term structures of interest rates;    -   generating, by means of said interest rate risk minimization        device: historical model errors; an optimal amount (Φ_(y)),        obtained as a function of said historical model errors, for        investment in each financial instrument of said second group of        financial instruments for protecting said first group of        financial instruments; and a residual risk estimation (E[Ψ_(t)        ²]); and        solving, by means of said interest rate risk minimization        device, equation:

${\frac{\partial{L\left( {\phi_{t},\mu_{t}} \right)}}{\partial\varphi_{y,t}} \approx \approx {{2{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{{\left\lbrack {\sum\limits_{k = 1}^{n}{c_{lk}D_{k}w_{y,k,t}}} \right\rbrack\left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left( {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right)}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{\theta_{k}^{2}c_{lk}^{2}D_{k}^{2}{w_{y,k,t}\left\lbrack {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right\rbrack}}}} \right\}}}} - \mu_{t}}} = 0$

-   -   to obtain μ_(t) and said optimal amount (Φ_(y)).        wherein D_(k) is a zero-coupon rate of maturity, C^(l) _(t)        represents the change in the l-th principal component between        time t and t+1, c_(lk) represents the sensitivity of the        zero-coupon rate of maturity D_(k) to this change, said C^(l)        _(t) and c_(lk) being the set of parameters of a term structure        model, w_(y,k,t) and w_(j,k,t), indicate the percentage of the        present value of bond y and j, respectively, represented by the        cash flow with maturity D_(k,) _(A) _(,k,t) is the value of the        liabilities included in the k-th time bucket amounts, μ_(t) is        the Lagrange multiplier and        wherein the ratio θ_(k), the volatility σ_(C) of the modeled        rate shifts and the model errors σ_(ε) are defined as

${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {{\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}}}}..}$

Another aim and other objects which will become better apparenthereinafter are achieved by a method for interest rate risk management,comprising:

-   -   receiving as input into an input device, a first group of data        about a first group of financial instruments to be protected and        a second group of data about a second group of financial        instruments protecting said first group of financial        instruments;    -   receiving as input into an interest rate risk minimization        device connected to said input device: said first and second        group of data; a data feed of current market prices of said        first and second group of financial instruments; a set of        parameters of a term structure model; and historical zero-coupon        term structures of interest rates;    -   generating, by means of said interest rate risk minimization        device: historical model errors; an optimal amount (Φ_(y)),        obtained as a function of said historical model errors, for        investment in each financial instrument of said second group of        financial instruments for protecting said first group of        financial instruments; and a residual risk estimation (E[Ψ_(t)        ²]); and        solving, by means of said interest rate risk minimization device        the following equation for each bond j included in the hedging        portfolio to obtain μ_(t) and said optimal amount (Φ_(y)):

$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$

wherein the factor F^(l) represents the l-th key zero rate change,c^(KRD) _(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to this change, ω_(y,k,t) and ω_(j,k,t), indicate thepercentage of the present value of bond y and j, respectively,represented by the cash flow with maturity D_(k,) _(A) _(,k,t) is thevalue of the liabilities included in the k-th time bucket amounts, μ_(t)is the Lagrange multiplier andthe volatility of the model errors σ_(ε) can be defined as in equation

${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$

Another aim and other objects which will become better apparenthereinafter are achieved by a method for interest rate risk management,comprising:

-   -   receiving as input into an input device, a first group of data        about a first group of financial instruments to be protected and        a second group of data about a second group of financial        instruments protecting said first group of financial        instruments;    -   receiving as input into an interest rate risk minimization        device connected to said input device: said first and second        group of data; a data feed of current market prices of said        first and second group of financial instruments; a set of        parameters of a term structure model; and historical zero-coupon        term structures of interest rates;    -   generating, by means of said interest rate risk minimization        device: historical model errors; an optimal amount (Φ_(y)),        obtained as a function of said historical model errors, for        investment in each financial instrument of said second group of        financial instruments for protecting said first group of        financial instruments; and a residual risk estimation (E[Ψ_(t)        ²]); and        solving, by means of said interest rate risk minimization device        the following equation for each bond j included in the hedging        portfolio to obtain μ_(t) and said optimal amount (Φ_(y)):

${2{\sum\limits_{k = 1}^{n}\left( {\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack} \right)}} = \mu_{t}$

wherein F^(l) represents the l-th risk factor change, D_(k) ^(X(l)-l)represents the sensitivity of the zero-coupon rate of maturity D_(k) tothis change, ω_(y,k,t) and ω_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and the volatility of the model errors σ_(ε) can be definedas in equation

$\begin{matrix}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} \\{= {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}}\end{matrix}$

BRIEF DESCRIPTION OF THE DRAWINGS

Further characteristics and advantages of the invention will becomebetter apparent from the detailed description of particular but notexclusive embodiments, illustrated by way of non-limiting examples inthe accompanying drawings, wherein:

FIG. 1 is block diagram depicting a system for interest rate riskmanagement in accordance with the invention.

FIG. 2 is block diagram depicting a preferred embodiment of theinvention, based on the PCA hedging model.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

A schematic diagram of a system for interest rate risk managementaccording to the present invention is shown in FIG. 1.

System 100 comprises an input device 200, an historical price database600, a bootstrapping engine 700, a historical zero-coupon term structureof interest rates 800, a term structure model estimation engine 900, aninterest rate risk minimization device 1200, and an output device 1800.

Input device 200 comprises any device suited to feed system 100 withdata, for instance a keyboard or a file reader. Data to be fed to system100 through input device 200 comprises a first group 300 of data aboutfinancial instruments contained in the balance sheet or portfolio to beprotected, a second group 400 of data that contains the informationrelated to the financial instruments which shall be used to protect thebalance sheet or portfolio.

The historical price database 600 may contain historical prices of thefinancial instruments underlying the term structure of interest rates.For example, for the US risk-free term structure, the CRSP Database forUS Treasuries can be used.

The bootstrapping engine 700 is a module that cooperates with thehistorical price database 600 in order to construct another datastructure, historical zero-coupon term structures of interest rates(including the current term structure) 800, starting from the historicalmarket prices of the financial instruments extracted by database 600.

The term structure model estimation engine 900 is a module thatestimates the parameters of a model explaining the dynamics of the termstructure of interest rates, relying on the historical zero-coupon termstructure of interest rates 800. Thus the term structure modelestimation engine 900 produces as output a set of parameters 1000 of amodel explaining the dynamics of the term structure of interest rates.

In a preferred embodiment this model relies on principal componentanalysis. However, the invention can also be applied to other modelsexplaining the dynamics of the term structure which are commonly usedfor minimizing interest rate risk, such as M-vector models, key rateduration models, generalized duration vector or duration vector.

The interest rate risk minimization device 1200 receives as input first300, and second 400 groups of data, historical zero-coupon termstructures of interest rates 800, parameters 1000 and current marketprices 1100 of the financial instruments, that is a data feed of currentmarket prices that allows to price the financial instruments which areinvolved in the current interest rate risk minimization problem.

The interest rate risk minimization device 1200 comprises an engine forestimation of historical model errors 1300; this engine, based on theinformation extracted from the historical zero-coupon term structures ofinterest rates 800 obtained by the bootstrapping engine 700 and the setof term structure parameters 1000, generates the model errors 1400; themodel errors are represented by the difference between actualzero-coupon interest rate changes and changes explained by the adoptedmodel of the term structure.

The interest rate risk minimization device 1200 also comprises aninterest rate risk minimization engine 1500, suited to minimize interestrate risk considering the model errors 1400 estimated by the modelerrors engine 1300, the set of term structure parameters 1000, andcurrent market prices 1100. The main minimization results are optimalweights 1600 for the financial instruments used to protect the balancesheet or portfolio against interest rate risk, and residual riskestimation 1700.

The calculated optimal weights 1600 indicate the optimal amount to beinvested in each financial instrument which shall be used to protect thebalance sheet or portfolio.

The residual risk estimation 1700 can be calculated based on the modelerrors 1400 estimated by the model errors engine 1300 and on thecalculated optimal weights 1600, and is an essential estimate in orderto determine if the considered strategy for minimizing interest raterisk is satisfactory or not.

The output device 1800 receives the optimal weights 1600, which can beused to generate buy and sell market orders 1900, and the residual riskestimation 1700 generated by the interest rate risk minimization device1200.

In a preferred embodiment, the invention is based on a PCA hedgingmodel. In this model the model error ε for a martingale zero-coupon rateof duration D_(k) can be defined as:

$\begin{matrix}{{{R\left( {t,D_{k}} \right)}} \equiv {{\sum\limits_{l = 1}^{M}{c_{lk}C_{t}^{l}}} + {ɛ\left( {t,D_{k}} \right)}}} & (1.)\end{matrix}$

where dR(t,D_(k)) represents an unexpected shift in the correspondingzero-coupon risk-free rate, C_(lt) represents the change in the l-thprincipal component between time t and t+1 and c_(lk) continues torepresent the sensitivity of the zero-coupon rate of maturity D_(k) tothis change.

In the hedging based on PCA of current state of art the expected hedgingerror due to the modeled behavior of interest rates is equal to zero, sothe error terms ε in equation (1.) are ignored; while in the PCA-hedgingof this invention the error terms ε in equation (1.) should beconsidered within the minimization of the expected immunization error.

The way in which the error terms ε in equation (1.) are consideredwithin the minimization can be simple but also sophisticated. Accordingto a simple approach, the volatility of the model errors σ_(ε) isassumed to be proportional to the volatility σ_(C) of the modeled rateshifts, so that their ratio θ_(k) is defined as:

$\begin{matrix}\begin{matrix}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} \\{= {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}}}}}\end{matrix} & (4.)\end{matrix}$

For the sake of simplicity, θ_(k) is assumed to be constant over time.

There is large empirical evidence that—for holding periods not longerthan one month—the effect of rate changes on the return provided by azero bond can be plausibly approximated by its duration D_(k). On thisbasis, it is possible to approximate the overall unexpected returnprovided by the combination of the two portfolios V and Has follows:

$\begin{matrix}{\psi_{t} \approx {{- {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}{\sum\limits_{k = 1}^{n}{{{R\left( {t,D_{k}} \right)}}D_{k}\omega_{y,k,t}}}}}} + {\sum\limits_{k = 1}^{m}{{{R\left( {t,D_{k}} \right)}}D_{k}A_{k,t}}}}} & (7.)\end{matrix}$

which, applying definition (1.), may be written as:

$\begin{matrix}{\psi_{t} \approx {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}{\sum\limits_{k = 1}^{n}{\left\lbrack {{\sum\limits_{l = 1}^{M}{c_{lk}C_{t}^{l}}} + {ɛ\left( {t,D_{k}} \right)}} \right\rbrack D_{k}\omega_{y,k,t}}}}} + {\sum\limits_{k = 1}^{m}{\left\lbrack {{\sum\limits_{l = 1}^{M}{c_{lk}C_{t}^{l}}} + {ɛ\left( {t,D_{k}} \right)}} \right\rbrack D_{A}A_{k,t}}}}} & (8.)\end{matrix}$

Since the residuals of a PCA have means equal to zero and areindependent from the principal components, the expected squared value ofthe unexpected return is:

$\begin{matrix}{{E\left\lbrack \psi_{t}^{2} \right\rbrack} \approx {E{\left\{ {\sum\limits_{l = 1}^{M}{C_{t}^{l}\left\lbrack {{\sum\limits_{k = 1}^{m}{c_{lk}D_{k}A_{k,t}}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}{\sum\limits_{k = 1}^{n}{c_{lk}D_{k}\omega_{y,k,t}}}}}} \right\rbrack}} \right\}^{2}++}E\left\{ {{\sum\limits_{k = 1}^{m}{{ɛ\left( {t,D_{k}} \right)}D_{k}A_{k,t}}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}{\sum\limits_{k = 1}^{n}{{ɛ\left( {t,D_{k}} \right)}D_{k}\omega_{y,k,t}}}}}} \right\}^{2}}} & (9.)\end{matrix}$

Assuming that the model error for a given rate k is independent from themodel errors for all other rates and applying the independence among theprincipal components as well as definition (4.), the last equationbecomes:

$\begin{matrix}{{E\left\lbrack \psi_{t}^{2} \right\rbrack} \approx {\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{\left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}}} \right\rbrack}}} \right\rbrack^{2}++}\left. \quad{\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{\theta_{k}^{2}c_{lk}^{2}{D_{k}^{2}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}}} \right\rbrack}^{2}}} \right\}} \right.}}} & (10.)\end{matrix}$

In order to minimize the equation (10.) subjecting to the self-financingconstraint, the first partial derivatives of the following Lagrangianfunction is set equal to zero, where μ_(t) is the Lagrange multiplier:

$\begin{matrix}{{L\left( {\phi_{t},\mu_{t}} \right)} = {{E\left( \psi_{t}^{2} \right)} - {\mu_{t}\left( {{\sum\limits_{y = 1}^{M + 1}\varphi_{y,t}} - H_{t}} \right)}}} & (11.)\end{matrix}$

Setting the first derivatives with respect to the amounts Φ_(y) equal tozero leads to following equation:

$\begin{matrix}{{\frac{\partial{L\left( {\phi_{t},\mu_{t}} \right)}}{\partial\varphi_{y,t}} \approx \approx {{2{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{{\left\lbrack {\sum\limits_{k = 1}^{n}{c_{lk}D_{k}\omega_{y,k,t}}} \right\rbrack \left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left( {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}\omega_{j,k,t}}} - A_{k,t}} \right)}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{\theta_{k}^{2}c_{lk}^{2}D_{k}^{2}{\omega_{y,k,t}\left\lbrack {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}\omega_{j,k,t}}} - A_{k,t}} \right\rbrack}}}} \right\}}}} - \mu_{t}}} = 0} & (12.)\end{matrix}$

This set of hedging equations must be applied to every y.

The set of hedging equations (12.) is subject to the self-financingconstraint (2.).

The PCA-hedging equation of the current state of art (5.) is a specialcase of the generalized hedging strategy (12.) of the present invention,specifically, it is the case assuming no model errors.

Accordingly, the improvement of PCA-hedging model presented in thisinvention within this particular embodiment consists in the term ofequations (12.) including θ_(k).

Within this term, θ_(k) represents the size of the expected model errorsfor rate R(t,Dk), whereas the exposure of the hedging strategy to theseerrors is represented by:

$\begin{matrix}{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}c_{lk}^{2}{D_{k}^{2}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}}} \right\rbrack}^{2}}} & (6.)\end{matrix}$

The purpose of the term of equations (12.) including θ_(k) is tointroduce a penalty for the exposure to model errors.

In this way the generalized PCA hedging implements a trade-off betweenthe precision of matching the sensitivity of portfolio V to eachprincipal component, that is the exclusive goal of traditionalPCA-hedging, and the level of exposure to model errors.

In a second preferred embodiment, the invention is based on a KRDhedging model. In this model the model error ε for a martingalezero-coupon rate of duration D_(k) can be defined as:

${{R\left( {t,D_{k}} \right)}} \equiv {{\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}}} + {ɛ\left( {t,D_{k}} \right)}}$

where in this case the factor F^(l) represents the l-th key zero ratechange and c^(KRD) _(lk) represents the sensitivity of the zero-couponrate of maturity D_(k) to this change which can be defined in differentways, for example, like in Nawalkha, Soto, and Beliaeva, Interest RateRisk Modeling. John Wiley & Sons, 2005.

Following the same approach described above for the PCA approach, theexpected squared value of the unexpected return can be approximated by:

${E_{t}\left\lbrack \psi_{t} \right\rbrack}^{2} \approx {{\sum\limits_{k = 1}^{n}{\sum\limits_{v = 1}^{n}{\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)\left( {{D_{v}A_{v,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{v}\omega_{y,k,t}}}} \right){E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}}} + {\sum\limits_{k = 1}^{n}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)^{2}}}}$

where the volatility of the model errors σ_(ε) can be defined as inequation (4.) or in a more sophisticated way. If we construct theLagrangian function as:

${L\left( {\phi_{t},\mu_{t}} \right)} = {{E\left( \psi_{t}^{2} \right)} - {\mu_{t}\left( {{\sum\limits_{y = 1}^{M + 1}\varphi_{y,t}} - H_{t}} \right)}}$

and we set its first derivatives equal to zero, we obtain theself-financing constraint and the following equations for each bond jincluded in the hedging portfolio:

$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$

The optimal weights φ_(y) to be invested in each bond can be calculatedbased on the last set of equations.

Yet in a third preferred embodiment, the invention is based on a GDVhedging model. In this model the model error □ for a martingalezero-coupon rate of duration Dk can be defined as:

${{dR}\left( {t,D_{k}} \right)} \equiv {{\sum\limits_{l = 1}^{M}\frac{F_{t}^{l}}{D_{k}^{1 - {\chi {(l)}}}}} + {ɛ\left( {t,D_{k}} \right)}}$

where in this case the factor F^(l) represents the l-th risk factorchange and D_(k) ^(X(l)-l) represents the sensitivity of the zero-couponrate of maturity D_(k) to this change.

Following the same approach described above for the PCA approach, theexpected squared value of the unexpected return can be approximated by:

${E_{t}\left\lbrack \psi_{t} \right\rbrack}^{2} \approx {\sum\limits_{k = 1}^{n}{\sum\limits_{v = 1}^{n}{\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)\left( {{D_{v}A_{v,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{v}\omega_{y,v,t}}}} \right){{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)^{2}}}}}}$

where the volatility of the model errors σ_(ε) can be defined as inequation (4.) or in a more sophisticated way. If we construct theLagrangian function as:

${L\left( {\phi_{t},\mu_{t}} \right)} = {{E\left( \psi_{t}^{2} \right)} - {\mu_{t}\left( {{\sum\limits_{y = 1}^{M + 1}\varphi_{y,t}} - H_{t}} \right)}}$

and we set its first derivatives equal to zero, we obtain theself-financing constraint and the following equations for each bond jincluded in the hedging portfolio:

${2{\sum\limits_{k = 1}^{n}\left( {\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack} \right)}} = \mu_{t}$

The optimal weights φ_(y) to be invested in each bond can be calculatedbased on the last set of equations.

A preferred embodiment of the invention based on PCA hedging model isshown in FIG. 2.

As in the general case depicted in FIG. 1, system 100 comprises an inputdevice 20 through which a user may insert a first group of data 21 aboutthe financial instruments contained in the balance sheet or portfolio tobe protected; this first group comprises the cash-flows CF1_(k) of thesefinancial instruments, the number m of time buckets in which the cashflow are grouped and the duration D_(k) of the time bucket.

A second group of data 22 is inserted, which is related to the financialinstruments y which shall be used to protect the balance sheet orportfolio; this second group comprises the number n of time buckets inwhich the cash flow of these financial instruments are grouped, thecash-flows CF2_(k) of these financial instruments, and the durationD_(k) of the time bucket.

Using the inserted values 21 and 22 and the data extracted from thehistorical price database 24, the bootstrapping engine 25 builds thehistorical zero coupon term structures 26.

Based on historical zero coupon term structures 26, the term structuremodel estimation engine 27 calculates the parameters 28 of a PCA modelexplaining the dynamics of the term structure of interest rates.Specifically, the obtained parameters 28 are the value C_(lt), thatrepresents the change in the l-th principal component between time t andt+1, and c_(lk), that represent the sensitivity of the zero-coupon rateof maturity D_(k) to this change.

In this example, the historical zero coupon term structures 26 and theparameters 28, C_(lt) and c_(lk), are sent as input to the historicalmodel errors engine 31; the historical model errors engine 31 isconfigured to solve the equation (4) and to produce as output the modelerrors 32, that contains the value of σ_(ε) and θ_(k).

Market prices of the financial instruments which shall be used toprotect the balance sheet or portfolio, P_(y) as well as of any listedsecurity which might be included in portfolio V, P_(z) are retrievedfrom market data providers in 33.

All the values inserted and updated, D_(k), m, n, M, CF1_(k), CF2_(k),P_(y), P_(z) and all calculated values C_(lt), c_(lk), θ_(k), are sentas input to the interest rate risk minimization engine 34, that relyingon the current zero-coupon term structure included in 26 firstcalculates A_(k), V as well as the percentage w_(y,k) of the presentvalue of bond y represented by the cash flow with maturity D_(k), andthen, using these calculated values, solves the equation 12 and producesas output the value 35 of the Lagrange multiplier μ_(t) and the optimalweights 36, that is Φ_(y).

The interest rate risk minimization engine 34 further uses the estimatedvalues 36 to solve the equation 10 and obtains as output the residualrisk estimation 37, that is E[Ψ_(t) ²].

The output device 38 receives the values Φy 36 and E[Ψ_(t) ²] 37 inorder to implement buy and sell orders 39 on the markets as in thegeneral case depicted in FIG. 1.

In this preferred embodiment the term structure model estimation engine900 is based on PCA hedging but, as shown in the general embodiment ofFIG. 1, this engine can be based also on alternative models, such asM-vector models, key rate duration models or duration vector models. Thesame logic used in this specific embodiment is maintained in the enginefor estimation of historical model errors 1300 while equations (10) and(12) in the interest rate risk management minimization engine 34 arederived, as already shown above, starting from a revised version ofequation (1) based on the adopted term structure model and thenfollowing the same steps shown above.

It has been shown that the invention fully achieves the intended aim andobjects, since it allows to adopt a generalized hedging model able tocontrol the overall exposure to the model errors and reduce residualinterest rate risk.

In particular, the interest rate risk minimization engine as disclosedin the present invention allows to control the exposure to model errorsin a hedging strategy.

This is a new feature that is ignored by current applications known inthe art for interest rate risk management.

An important result obtained by a system according to this invention isan average reduction in the hedging errors of 35%.

In the particular case of the PCA hedging model, the generalized modelproposed in this invention permits to the 3-component PCA to outperform2-component PCA. Similar advantages can also be expected for the KRD andGDV models.

Another advantage obtained by a system according to this invention isthe estimation of the residual risk in order to determine if theconsidered strategy for minimizing interest rate risk is satisfactory ornot. This is another new feature of this invention ignored by currentapplications minimizing interest rate risk which assume residual risk tobe zero.

Clearly, several modifications will be apparent to and can be readilymade by the skilled in the art without departing from the scope of thepresent invention.

Therefore, the scope of the claims shall not be limited by theillustrations or the preferred embodiments given in the description inthe form of examples, but rather the claims shall encompass all of thefeatures of patentable novelty that reside in the present invention,including all the features that would be treated as equivalents by theskilled in the art.

What is claimed is:
 1. A system for interest rate risk management,comprising: an input device, configured to receive as input a firstgroup of data indicative of a first group of financial instruments to beprotected; a second group of data indicative of a second group offinancial instruments for protecting said first group of financialinstruments; and an interest rate risk minimization device, connected tosaid input device, configured to receive as input said first and secondgroup of data, a data feed of current market prices of said first andsecond group of financial instruments, a set of parameters of a termstructure model, and historical zero-coupon termstructures of interestrates; wherein said interest rate risk minimization device is furtherconfigured to generate: historical model errors; an optimal amount(Φ_(y)), obtained as a function of said historical model errors, forinvestment in each financial instrument of said second group offinancial instruments for protecting said first group of financialinstruments; and a residual risk estimation (E[Ψ_(t) ²]); wherein saidinterest rate risk minimization device is further configured to solveequation:${\frac{\partial{L\left( {\phi_{t},\mu_{t}} \right)}}{\partial\varphi_{y,t}} \approx \approx {{2{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\begin{Bmatrix}{{\left\lbrack {\sum\limits_{k = 1}^{n}{c_{lk}D_{k}w_{y,k,t}}} \right\rbrack\left\lbrack {\sum\limits_{k = 1}^{m\; {{ax}{\lbrack{m,n}\rbrack}}}{c_{lk}{D_{k}\left( {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right)}}} \right\rbrack}++} \\{\sum\limits_{k = 1}^{n}{\theta_{k}^{2}c_{lk}^{2}D_{k}^{2}{w_{y,k,t}\left\lbrack {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right\rbrack}}}\end{Bmatrix}}}} - \mu_{t}}} = 0$ to obtain μ_(t) and said optimalamount (Φ_(y)), wherein D_(k) is a zero-coupon rate of maturity, C^(l)_(t) represents the change in the l-th principal component between timet and t+1, c_(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to this change, said C^(l) _(t) and c_(lk) being the setof parameters of a term structure model, w_(y,k,t) and w_(j,k,t),indicate the percentage of the present value of bond y and j,respectively, represented by the cash flow with maturity D_(k,) _(A)_(,k,t) is the value of the liabilities included in the k-th time bucketamounts, μ_(t) is the Lagrange multiplier and wherein the ratio θ_(k),the volatility σ_(C) of the modeled rate shifts and the model errorsσ_(ε) are defined as${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$2. A system for interest rate risk management, comprising: an inputdevice, configured to receive as input a first group of data indicativeof a first group of financial instruments to be protected; a secondgroup of data indicative of a second group of financial instruments forprotecting said first group of financial instruments; and an interestrate risk minimization device, connected to said input device,configured to receive as input said first and second group of data, adata feed of current market prices of said first and second group offinancial instruments, a set of parameters of a term structure model,and historical zero-coupon termstructures of interest rates; whereinsaid interest rate risk minimization device is further configured togenerate: historical model errors; an optimal amount (Φ_(y)), obtainedas a function of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); wherein said interest rate riskminimization device is further configured to solve the followingequation for each bond j included in the hedging portfolio to obtainμ_(t) and said optimal amount (Φ_(y)):$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$wherein the factor F^(l) represents the l-th key zero rate change,c^(KRD) _(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to this change, ω_(y,k,t) and ω_(j,k,t), indicate thepercentage of the present value of bond y and j, respectively,represented by the cash flow with maturity D_(k,) _(A) _(,k,t) is thevalue of the liabilities included in the k-th time bucket amounts, μ_(t)is the Lagrange multiplier and the volatility of the model errors σ_(ε)can be defined as in equation${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$3. A system for interest rate risk management, comprising: an inputdevice, configured to receive as input a first group of data indicativeof a first group of financial instruments to be protected; a secondgroup of data indicative of a second group of financial instruments forprotecting said first group of financial instruments; and an interestrate risk minimization device, connected to said input device,configured to receive as input said first and second group of data, adata feed of current market prices of said first and second group offinancial instruments, a set of parameters of a term structure model,and historical zero-coupon termstructures of interest rates; whereinsaid interest rate risk minimization device is further configured togenerate: historical model errors; an optimal amount (Φ_(y)), obtainedas a function of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); wherein said interest rate riskminimization device is further configured to solve the followingequation for each bond j included in the hedging portfolio to obtainμ_(t) and said optimal amount (Φ_(y)):${2{\sum\limits_{k = 1}^{n}\left( {\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack} \right)}} = \mu_{t}$wherein F^(l) represents the l-th risk factor change, D_(k) ^(x(l)-l)represents the sensitivity of the zero-coupon rate of maturity D_(k) tothis change, ω_(y,k,t) and ω_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and the volatility of the model errors σ_(ε) can be definedas in equation${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}}}}$4. The system according to claim 1, 2 or 3 wherein said interest raterisk minimization device comprises an engine for estimation of saidhistorical model errors, configured to receive as input said a set ofparameters of a term structure model and information extracted from saidhistorical zero-coupon term structures of interest rates and to generatesaid historical model errors.
 5. The system according to claim 4,wherein said interest rate risk minimization device comprises aninterest rate risk minimization engine, configured to receive as inputsaid first group of data indicative of a first group of financialinstruments to be protected, said second group of data indicative of asecond group of protecting financial instruments, said historical modelerrors, the set of parameters of a term structure model, said historicalzero-coupon term structures of interest rates, and said current marketprices and to generate said optimal amount (Φ_(y)) for investment ineach financial instrument of said second group of financial instrumentsfor protecting said first group of financial instruments, and saidresidual risk estimation (E[Ψ_(t) ²]).
 6. The system according to claim5, further comprising a historical price database, storing historicalprices of financial instruments underlying the historical zero-couponterm structures of interest rates, and a bootstrapping engine, connectedto said input device and to said historical price database, configuredto generate said historical zero-coupon term structures of interestrates.
 7. The system according to claim 6, further comprising a termstructure model estimation engine, receiving said historical zero-couponterm structures of interest rates, and configured to generate said setof parameters of a term structure model.
 8. The system according claim7, further comprising an output device, configured to receive as inputsaid optimal amount for investment in each financial instrument of saidsecond group of financial instruments for generating buy and sell marketorders, and said residual risk estimation.
 9. The system according toclaim 8, wherein said term structure model relies on Principal ComponentAnalysis.
 10. The system according to claim 1, wherein said interestrate risk minimization device is further configured to solve anequation:${E\left\lbrack \psi_{t}^{2} \right\rbrack} \approx {\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{\left\lbrack {\sum\limits_{k = 1}^{m\; {{ax}{\lbrack{m,n}\rbrack}}}{c_{lk}{D_{k}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}w_{y,k,t}}}} \right\rbrack}}} \right\rbrack^{2}++}{\sum\limits_{k = 1}^{m\; {{ax}{\lbrack{m,n}\rbrack}}}{\theta_{k}^{2}c_{lk}^{2}{D_{k}^{2}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}w_{y,k,t}}}} \right\rbrack}^{2}}}} \right\}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]).
 11. The systemaccording to claim 2 wherein said interest rate risk minimization deviceis further configured to solve an equation:${E_{t}\left\lbrack \psi_{t} \right\rbrack}^{2} \approx {\sum\limits_{k = 1}^{n}{\sum\limits_{v = 1}^{n}{\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)\left( {{D_{v}A_{v,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{v}\omega_{y,v,t}}}} \right){{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)^{2}}}}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]).
 12. The systemaccording to claim 3, wherein said interest rate risk minimizationdevice is further configured to solve an equation:${E_{t}\left\lbrack \psi_{t} \right\rbrack}^{2} \approx {\sum\limits_{k = 1}^{n}{\sum\limits_{v = 1}^{n}{\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)\left( {{D_{v}A_{v,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{v}\omega_{y,v,t}}}} \right){{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)^{2}}}}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]).
 13. A method forinterest rate risk management, comprising: receiving as input into aninput device, a first group of data about a first group of financialinstruments to be protected and a second group of data about a secondgroup of financial instruments protecting said first group of financialinstruments; receiving as input into an interest rate risk minimizationdevice connected to said input device: said first and second group ofdata; a data feed of current market prices of said first and secondgroup of financial instruments; a set of parameters of a term structuremodel; and historical zero-coupon term structures of interest rates;generating, by means of said interest rate risk minimization device:historical model errors; an optimal amount (Φ_(y)), obtained as afunction of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); and solving, by means of said interestrate risk minimization device, equation:${\frac{\partial{L\left( {\phi_{t},\mu_{t}} \right)}}{\partial\varphi_{y,t}} \approx \approx {{2{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{{\left\lbrack {\sum\limits_{k = 1}^{n}{c_{lk}D_{k}w_{y,k,t}}} \right\rbrack \left\lbrack {\sum\limits_{k = 1}^{m\; {{ax}{\lbrack{m,n}\rbrack}}}{c_{lk}{D_{k}\left( {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right)}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{\theta_{k}^{2}c_{lk}^{2}D_{k}^{2}{w_{y,k,t}\left\lbrack {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right\rbrack}}}} \right\}}}} - \mu_{t}}} = 0$to obtain μ_(t) and said optimal amount (Φ_(y)), wherein D_(k) is azero-coupon rate of maturity, C^(l) _(t) represents the change in thel-th principal component between time t and t+1, c_(lk) represents thesensitivity of the zero-coupon rate of maturity D_(k) to this change,said C^(l) _(t) and c_(lk) being the set of parameters of a termstructure model, w_(y,k,t) and w_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and wherein the ratio θ_(k), the volatility σ_(C) of themodeled rate shifts and the model errors σ_(ε) are defined as${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$14. A method for interest rate risk management, comprising: receiving asinput into an input device, a first group of data about a first group offinancial instruments to be protected and a second group of data about asecond group of financial instruments protecting said first group offinancial instruments; receiving as input into an interest rate riskminimization device connected to said input device: said first andsecond group of data; a data feed of current market prices of said firstand second group of financial instruments; a set of parameters of a termstructure model; and historical zero-coupon term structures of interestrates; generating, by means of said interest rate risk minimizationdevice: historical model errors; an optimal amount (Φ_(y)), obtained asa function of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); and solving, by means of said interestrate risk minimization device the following equation for each bond jincluded in the hedging portfolio to obtain μ_(t) and said optimalamount (Φ_(y)):$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$wherein the factor F^(l) represents the l-th key zero rate change,c^(KRD) _(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to this change, ω_(y,k,t) and ω_(j,k,t), indicate thepercentage of the present value of bond y and j, respectively,represented by the cash flow with maturity D_(k,) _(A) _(,k,t) is thevalue of the liabilities included in the k-th time bucket amounts, μ_(t)is the Lagrange multiplier and the volatility of the model errors σ_(ε)can be defined as in equation${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}}}}$15. A method for interest rate risk management, comprising: receiving asinput into an input device, a first group of data about a first group offinancial instruments to be protected and a second group of data about asecond group of financial instruments protecting said first group offinancial instruments; receiving as input into an interest rate riskminimization device connected to said input device: said first andsecond group of data; a data feed of current market prices of said firstand second group of financial instruments; a set of parameters of a termstructure model; and historical zero-coupon term structures of interestrates; generating, by means of said interest rate risk minimizationdevice: historical model errors; an optimal amount (Φ_(y)), obtained asa function of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); and solving, by means of said interestrate risk minimization device the following equation for each bond jincluded in the hedging portfolio to obtain μ_(t) and said optimalamount (Φ_(y)):$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$wherein F^(l) represents the l-th risk factor change, D_(k) ^(X(l)-l)represents the sensitivity of the zero-coupon rate of maturity D_(k) tothis change, ω_(y,k,t) and ω_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and the volatility of the model errors σ_(ε) can be definedas in equation${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$16. The method according to claim 13, 14 or 15, further comprisingstoring, in a historical price database, historical prices of financialinstruments underlying the historical zero-coupon term structures ofinterest rates; and generating, by a bootstrapping engine, connected tosaid input device and to said historical price database, said historicalzero-coupon term structures of interest rates.
 17. The method accordingto claim 16, further comprising generating, by a term structure modelestimation engine, receiving said historical zero-coupon term structureof interest rates, said set of parameters of a term structure model. 18.The method according to claim 17, wherein said step of generatinghistorical model errors comprises receiving as input, by an engineadapted to estimate historical model errors comprised in said interestrate risk minimization device, said set of parameters of a termstructure model and information extracted from said historicalzero-coupon term structures of interest rates.
 19. The method accordingto claim 18, wherein said step of generating said residual riskestimation (E[Ψ_(t) ²]) comprises receiving as input, by an interestrate risk minimization engine comprised in said interest rate riskminimization device, said historical model errors and generating saidoptimal amount (Φ_(y)) for investment in each financial instrument ofsaid second group of financial instruments.
 20. A system for interestrate risk management, comprising: an input device, configured to receiveas input a first group of data indicative of a first group of financialinstruments to be protected; a second group of data indicative of asecond group of financial instruments for protecting said first group offinancial instruments; and an interest rate risk minimization device,connected to said input device, configured to receive as input saidfirst and second group of data, a data feed of current market prices ofsaid first and second group of financial instruments, a set ofparameters of a term structure model, and historical zero-coupon termstructures of interest rates; wherein said interest rate riskminimization device is further configured to generate: historical modelerrors; an optimal amount (Φ_(y)), obtained as a function of saidhistorical model errors, for investment in each financial instrument ofsaid second group of financial instruments for protecting said firstgroup of financial instruments; and a residual risk estimation; whereinsaid interest rate risk minimization device comprises an engine forestimation of historical model errors which is configured to solve anequation:${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}}}}$to obtain θ_(k). wherein θ_(k) is a ratio, the σ_(C) is the volatilitythe modeled rate shifts, σ_(ε) is a model error, C^(l) _(t) representsthe change in the l-th principal component between time t and t+1,c_(lk) represents the sensitivity of the zero-coupon rate of maturityD_(k) to this change, said C^(l) _(t) and c_(lk) being the set ofparameters of a term structure model.
 21. A system for interest raterisk management, comprising: an input device, configured to receive asinput a first group of data indicative of a first group of financialinstruments to be protected; a second group of data indicative of asecond group of financial instruments for protecting said first group offinancial instruments; and an interest rate risk minimization device,connected to said input device, configured to receive as input saidfirst and second group of data, a data feed of current market prices ofsaid first and second group of financial instruments, a set ofparameters of a term structure model, and historical zero-coupon termstructures of interest rates; wherein said interest rate riskminimization device is further configured to generate: historical modelerrors; an optimal amount (Φ_(y)), obtained as a function of saidhistorical model errors, for investment in each financial instrument ofsaid second group of financial instruments for protecting said firstgroup of financial instruments; and a residual risk estimation (E[Ψ_(t)²]); wherein said interest rate risk minimization device is configuredto solve an equation:${E\left\lbrack \psi_{t}^{2} \right\rbrack} \approx {\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{\left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}w_{y,k,t}}}} \right\rbrack}}} \right\rbrack^{2}++}{\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{\theta_{k}^{2}c_{lk}^{2}{D_{k}^{2}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}w_{y,k,t}}}} \right\rbrack}^{2}}}} \right\}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]), wherein saidinterest rate risk minimization device is further configured to solve anequation:$\frac{\partial{L\left( {\phi_{t},\mu_{t}} \right)}}{\partial\varphi_{y,t}} \approx \approx {2{\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{{\left\lbrack {\sum\limits_{k = 1}^{n}{c_{lk}D_{k}w_{y,k,t}}} \right\rbrack \left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left( {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right)}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{\theta_{k}^{2}c_{lk}^{2}D_{k}^{2}{w_{y,k,t}\left\lbrack {{\sum\limits_{j = 1}^{M + 1}{\varphi_{j,t}w_{j,k,t}}} - A_{k,t}} \right\rbrack}}}} \right\}}}}$to obtain μ_(t) and said optimal amount (Φ_(y)), wherein D_(k) is azero-coupon rate of maturity, C^(l) _(t) represents the change in thel-th principal component between time t and t+1, c_(lk) represents thesensitivity of the zero-coupon rate of maturity D_(k) to this change,said C^(l) _(t) and c_(lk) being the set of parameters of a termstructure model, w_(y,k,t) and w_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and wherein the ratio θ_(k), the volatility σ_(C) of themodeled rate shifts and the model errors σ_(ε) are defined as${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$22. A system for interest rate risk management, comprising: an inputdevice, configured to receive as input a first group of data indicativeof a first group of financial instruments to be protected; a secondgroup of data indicative of a second group of financial instruments forprotecting said first group of financial instruments; and an interestrate risk minimization device, connected to said input device,configured to receive as input said first and second group of data, adata feed of current market prices of said first and second group offinancial instruments, a set of parameters of a term structure model,and historical zero-coupon term structures of interest rates; whereinsaid interest rate risk minimization device is further configured togenerate: historical model errors; an optimal amount (Φ_(y)), obtainedas a function of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); wherein said interest rate riskminimization device is further configured to solve the followingequation for each bond j included in the hedging portfolio to obtainμ_(t) and said optimal amount (Φ_(y)):$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$wherein the factor F^(l) represents the l-th key zero rate change,c^(KRD) _(lk) represents the sensitivity of the zero-coupon rate ofmaturity D_(k) to this change, ω_(y,k,t) and ω_(j,k,t), indicate thepercentage of the present value of bond y and j, respectively,represented by the cash flow with maturity D_(k,) _(A) _(,k,t) is thevalue of the liabilities included in the k-th time bucket amounts, μ_(t)is the Lagrange multiplier and the volatility of the model errors σ_(ε)can be defined as in equation${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}.}}}}$23. A system for interest rate risk management, comprising: an inputdevice, configured to receive as input a first group of data indicativeof a first group of financial instruments to be protected; a secondgroup of data indicative of a second group of financial instruments forprotecting said first group of financial instruments; and an interestrate risk minimization device, connected to said input device,configured to receive as input said first and second group of data, adata feed of current market prices of said first and second group offinancial instruments, a set of parameters of a term structure model,and historical zero-coupon term structures of interest rates; whereinsaid interest rate risk minimization device is further configured togenerate: historical model errors; an optimal amount (Φ_(y)), obtainedas a function of said historical model errors, for investment in eachfinancial instrument of said second group of financial instruments forprotecting said first group of financial instruments; and a residualrisk estimation (E[Ψ_(t) ²]); wherein said interest rate riskminimization device is further configured to solve the followingequation for each bond j included in the hedging portfolio to obtainμ_(t) and said optimal amount (Φ_(y)):$2{\sum\limits_{k = 1}^{n}\left( {{\left\lbrack {{\sum\limits_{v = 1}^{n}{\omega_{j,v,t}D_{v}D_{k}{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}}} + {{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}D_{k}^{2}\omega_{j,k,t}}} \right\rbrack \left. \quad\left\lbrack {{\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}\omega_{y,k,t}}} - A_{k,t}} \right\rbrack \right)} = \mu_{t}} \right.}$wherein F^(l) represents the l-th risk factor change, D_(k) ^(X(l)-l)represents the sensitivity of the zero-coupon rate of maturity D_(k) tothis change, ω_(y,k,t) and ω_(j,k,t), indicate the percentage of thepresent value of bond y and j, respectively, represented by the cashflow with maturity D_(k,) _(A) _(,k,t) is the value of the liabilitiesincluded in the k-th time bucket amounts, μ_(t) is the Lagrangemultiplier and the volatility of the model errors σ_(ε) can be definedas in equation${{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2} \equiv {\theta_{k}^{2}{\sigma_{C}\left( {t,D_{k}} \right)}^{2}}} = {\theta_{k}^{2}{\sum\limits_{l = 1}^{M}{c_{lk}^{2}{E\left\lbrack C_{t}^{l} \right\rbrack}^{2}}}}$24. The method according to claim 13, further comprising solving, bymeans of said interest rate risk minimization device, an equation:${E\left\lbrack \psi_{t}^{2} \right\rbrack} \approx {\sum\limits_{l = 1}^{M}{{E\left\lbrack \left( C_{t}^{l} \right)^{2} \right\rbrack}\left\{ {{\left\lbrack {\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{c_{lk}{D_{k}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}w_{y,k,t}}}} \right\rbrack}}} \right\rbrack^{2}++}{\sum\limits_{k = 1}^{\max {\lbrack{m,n}\rbrack}}{\theta_{k}^{2}c_{lk}^{2}{D_{k}^{2}\left\lbrack {A_{k,t} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}w_{y,k,t}}}} \right\rbrack}^{2}}}} \right\}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]).
 25. The methodaccording to claim 14, further comprising solving, by means of saidinterest rate risk minimization device, an equation:${E_{t}\left\lbrack \psi_{t} \right\rbrack}^{2} \approx {\sum\limits_{k = 1}^{n}{\sum\limits_{v = 1}^{n}{\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)\left( {{D_{v}A_{v,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{v}\omega_{y,v,t}}}} \right){{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{c_{lk}^{KRD}F_{t}^{l}{\sum\limits_{h = 1}^{M}{c_{hv}^{KRD}F_{t}^{h}}}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)^{2}}}}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]).
 26. The methodaccording to claim 15, further comprising solving, by means of saidinterest rate risk minimization device, an equation:${E_{t}\left\lbrack \psi_{t} \right\rbrack}^{2} \approx {\sum\limits_{k = 1}^{n}{\sum\limits_{v = 1}^{n}{\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)\left( {{D_{v}A_{v,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{v}\omega_{y,v,t}}}} \right){{E_{t}\left\lbrack {\sum\limits_{l = 1}^{M}{D_{k}^{{\chi {(l)}} - 1}F_{t}^{l}{\sum\limits_{h = 1}^{M}{D_{v}^{{\chi {(h)}} - 1}F_{t}^{h}}}}} \right\rbrack}++}{\sum\limits_{k = 1}^{n}{{\sigma_{ɛ}\left( {t,D_{k}} \right)}^{2}\left( {{D_{k}A_{k,t}} - {\sum\limits_{y = 1}^{M + 1}{\varphi_{y,t}D_{k}\omega_{y,k,t}}}} \right)^{2}}}}}}$to obtain said residual risk estimation (E[Ψ_(t) ²]).